Common Root of Quadratic Equations in G.P.

To find the conditions for the two quadratic equations to have a common root, we need to establish a relationship between the coefficients of the equations. Let's consider the given equations:

Equation 1: ax^2 + bx + c = 0
Equation 2: dx^2 + 2ex + f = 0

Step 1: Identifying the Common Root
For the two equations to have a common root, the discriminant (b^2 - 4ac) of both equations must be equal to zero. This condition ensures that both equations have the same root.

Step 2: Finding the Discriminant for Equation 1
The discriminant for Equation 1 is given by:
D1 = b^2 - 4ac

Step 3: Finding the Discriminant for Equation 2
The discriminant for Equation 2 is given by:
D2 = (2e)^2 - 4d(f)

Step 4: Equating the Discriminants
For the two equations to have a common root, the discriminants must be equal:
D1 = D2

Step 5: Simplifying the Equation
Substituting the values of D1 and D2, we have:
b^2 - 4ac = (2e)^2 - 4d(f)

Step 6: Rearranging the Equation
Rearranging the equation, we get:
4d(f) + 4ac = 4e^2 + b^2

Step 7: Rearranging the Equation Further
Dividing both sides by 4 gives us:
df + ac = e^2 + (b^2)/4

Step 8: Simplifying the Equation
We can rewrite the equation as:
df + ac = e^2 + (b^2)/4

Step 9: Observing the Relationship
From the equation, we can observe that the left-hand side (df + ac) is the sum of the product of the first and third terms of the G.P. (a and c) and the product of the second term (b/2) and the geometric mean (b/2). The right-hand side (e^2 + (b^2)/4) is the sum of the squares of the second and third terms of the G.P. (b/2 and c).

Therefore, we can conclude that if d/a, e/b, and f/c are in geometric progression (G.P.), then the quadratic equations ax^2 + bx + c = 0 and dx^2 + 2ex + f = 0 have a common root.